diff --git a/docs/api-completeness/index.md b/docs/api-completeness/index.md
index 3daba31ca..331817920 100644
--- a/docs/api-completeness/index.md
+++ b/docs/api-completeness/index.md
@@ -1,10 +1,5 @@
 # API Completeness
 
-Narwhals has two different level of support for libraries: "full" and "interchange".
-
-Libraries for which we have full support we intend to support the whole Narwhals API,
-however this is a continuous work in progress.
-
 In the following section it is possible to check which method is implemented for which
 class and backend.
 
@@ -12,3 +7,4 @@ class and backend.
 
     - By design, Polars supports all the methods of the Narwhals API.
     - "pandas-like" means pandas, cuDF and Modin.
+    - "spark-like" means PySpark, but we may extend this to also cover SQLFrame.
diff --git a/narwhals/functions.py b/narwhals/functions.py
index a8d098a39..f10a397f7 100644
--- a/narwhals/functions.py
+++ b/narwhals/functions.py
@@ -10,6 +10,7 @@
 from typing import Sequence
 from typing import TypeVar
 from typing import Union
+from typing import overload
 
 from narwhals._expression_parsing import extract_compliant
 from narwhals._pandas_like.utils import broadcast_align_and_extract_native
@@ -38,7 +39,9 @@
     from narwhals.dtypes import DType
     from narwhals.schema import Schema
     from narwhals.series import Series
+    from narwhals.typing import IntoDataFrameT
     from narwhals.typing import IntoExpr
+    from narwhals.typing import IntoFrameT
     from narwhals.typing import IntoSeriesT
 
     class ArrowStreamExportable(Protocol):
@@ -47,11 +50,35 @@ def __arrow_c_stream__(
         ) -> object: ...
 
 
+@overload
 def concat(
-    items: Iterable[FrameT],
+    items: Iterable[DataFrame[IntoDataFrameT]],
     *,
     how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
-) -> FrameT:
+) -> DataFrame[IntoDataFrameT]: ...
+
+
+@overload
+def concat(
+    items: Iterable[LazyFrame[IntoFrameT]],
+    *,
+    how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
+) -> LazyFrame[IntoFrameT]: ...
+
+
+@overload
+def concat(
+    items: Iterable[DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]],
+    *,
+    how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
+) -> DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]: ...
+
+
+def concat(
+    items: Iterable[DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]],
+    *,
+    how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
+) -> DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]:
     """Concatenate multiple DataFrames, LazyFrames into a single entity.
 
     Arguments:
@@ -197,7 +224,7 @@ def concat(
     validate_laziness(items)
     first_item = items[0]
     plx = first_item.__narwhals_namespace__()
-    return first_item._from_compliant_dataframe(  # type: ignore[return-value]
+    return first_item._from_compliant_dataframe(
         plx.concat([df._compliant_frame for df in items], how=how),
     )
 
diff --git a/narwhals/stable/v1/__init__.py b/narwhals/stable/v1/__init__.py
index 689ab53dd..e8fd5a921 100644
--- a/narwhals/stable/v1/__init__.py
+++ b/narwhals/stable/v1/__init__.py
@@ -3292,25 +3292,33 @@ def max_horizontal(*exprs: IntoExpr | Iterable[IntoExpr]) -> Expr:
 
 @overload
 def concat(
-    items: Iterable[DataFrame[Any]],
+    items: Iterable[DataFrame[IntoDataFrameT]],
     *,
     how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
-) -> DataFrame[Any]: ...
+) -> DataFrame[IntoDataFrameT]: ...
+
+
+@overload
+def concat(
+    items: Iterable[LazyFrame[IntoFrameT]],
+    *,
+    how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
+) -> LazyFrame[IntoFrameT]: ...
 
 
 @overload
 def concat(
-    items: Iterable[LazyFrame[Any]],
+    items: Iterable[DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]],
     *,
     how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
-) -> LazyFrame[Any]: ...
+) -> DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]: ...
 
 
 def concat(
-    items: Iterable[DataFrame[Any] | LazyFrame[Any]],
+    items: Iterable[DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]],
     *,
     how: Literal["horizontal", "vertical", "diagonal"] = "vertical",
-) -> DataFrame[Any] | LazyFrame[Any]:
+) -> DataFrame[IntoDataFrameT] | LazyFrame[IntoFrameT]:
     """Concatenate multiple DataFrames, LazyFrames into a single entity.
 
     Arguments:
@@ -3446,7 +3454,7 @@ def concat(
         │ 4   ┆ null ┆ y    │
         └─────┴──────┴──────┘
     """
-    return _stableify(nw.concat(items, how=how))  # type: ignore[no-any-return]
+    return _stableify(nw.concat(items, how=how))
 
 
 def concat_str(
diff --git a/tests/frame/invalid_test.py b/tests/frame/invalid_test.py
index af600321a..5d2f7b45a 100644
--- a/tests/frame/invalid_test.py
+++ b/tests/frame/invalid_test.py
@@ -40,7 +40,7 @@ def test_validate_laziness() -> None:
         TypeError,
         match=("The items to concatenate should either all be eager, or all lazy"),
     ):
-        nw.concat([nw.from_native(df, eager_only=True), nw.from_native(df).lazy()])  # type: ignore[list-item]
+        nw.concat([nw.from_native(df, eager_only=True), nw.from_native(df).lazy()])
 
 
 @pytest.mark.slow